Creation and Annihilation

Creation and annihilation operators are operators that create or destroy a particle. We interpret ${\hat{a}}^{†}$ as a creation operator, and $\hat{a}$ as a annihilation operator. They are found by solving the Quantum Harmonic Oscillator.

The creation and annihilation operators acting on the Fock basis are given by

$$\hat{a}|n⟩=\sqrt{n}|n-1⟩,{\hat{a}}^{†}\sqrt{n+1}|n+1⟩.$$

For bosons, the creation and annihilation operators obey a commutation identity,

$$[\hat{a},{\hat{a}}^{†}]=1,[\hat{N},{\hat{a}}^{†}]={\hat{a}}^{†},[\hat{N},\hat{a}]=-\hat{a},$$

For fermions, they obey an anti-commutation identity:

$$\{\hat{c},{\hat{c}}^{†}\}=1,\{\hat{N},{\hat{a}}^{†}\}={\hat{c}}^{†},\{\hat{N},\hat{c}\}=-\hat{c},$$